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New Haven, Connecticut 


Introduction and Summary 655 

Stress-Differences between Contiguous Columns of the Crust 659 

Stresses under Conditions of Isostatic Equilibrium .... 659 

Modifications of Stresses Produced by Base-Leveling . . . 666 

Relief of Stress Accompanying Restoration of Isostasy . . . 670 

Relations of Undertow to the Zone of Compensation . . 672 

Present Status of the Problem 672 

Objections against Undertow in the Zone of Compensation . . 677 

Undertow Restricted to a Sphere of Weakness — the Asthenosphere 680 


In studies on the nature of isostasy it is necessary to distinguish 
between, first, the existence of isostasy; second, the limits of 
isostatic equilibrium; and third, the mode of maintenance of this 

The first has long been known, the knowledge of the existence 
of some relation of density counterbalancing elevation having been 
gradually developed since the middle of the nineteenth century 
through the determination of the local deviations of the vertical 
as shown by the comparison of the astronomic and geodetic latitudes 
for the same station. This was a problem which arose in both 
astronomy and geodesy. It was found, when the attractive effect 
of the mountain regions was computed, that they did not deflect 
the vertical at adjacent stations as much as was to be expected 
from their visible masses. The phenomenon was first pointed out 

1 An abstract of Parts V, VI, and VII of this series was given at the April, 1914, 
meeting of the American Philosophical Society at Philadelphia under the title, 
"Relations of Isostasy to a Zone of Weakness — the Asthenosphere." See Science, 
XXXLX, 842. 



by Petit in 1849. 1 Archdeacon Pratt of Calcutta showed a few 
years later that whereas a discrepancy of 5 . 2" existed between the 
geodetic and astronomic latitudes of Kalianpur and Kaliana, the 
calculation of the effect of the Himalayas called for a difference of 

These facts were definitely formulated into a theory of isostasy 
by the Astronomer Royal of Great Britain, G. B. Airy, within a 
year following the appearance of Pratt's paper, 3 though it remained 
for Dutton to recognize the large geologic significance and to coin 
for the relations of elevation and density the word isostasy. 4 Fol- 
lowing this Putnam and Gilbert showed by gravity measurements 
that a considerable degree of regional isostasy existed over the 
United States. 5 Since then has appeared the much more detailed 
work of Hayford and Bowie, the computations made by the comput- 
ing office of the United States Coast and Geodetic Survey under 
their directions making possible this present investigation. 

Thus there has developed through more than half a century 
evidence beyond controversy which shows that the earth's crust 
in its larger relief and, within certain limits, even its smaller features, 
such as the great plateaus and basins, rests more or less approxi- 
mately in flotational equilibrium. 

The second division of the larger problem of isostasy, that of the 
areal limits and degree of perfection of isostatic adjustment, is the 
subject which has been dealt with in the previous parts of this 
investigation. It has been found that, although the relations of 
continents and ocean basins show with respect to each other a high 

1 " Sur la latitude de l'Observatoire de Toulouse, la density moyenne de la Chaine 
des Pyr6nees, et la probability qu'il existe un vide sons cette chaine," Comples rendus 
de I'Acad. des Sc, XXIX (1849), 73°- 

2 "On the Attraction of the Himalaya Mountains and of the Elevated Regions 
beyond Them, upon the Plumbline in India," Phil. Trans. Roy. Soc, Vol. CXLV 

> G. B. Airy, "On the Computation of the Effect of the Attraction of Mountain 
Masses as Disturbing the Apparent Astronomical Latitude of Stations in Geodetic 
Surveys," Phil. Trans. Roy. Soc, Vol. CXLV (1853). 

< "On Some of the Greater Problems of Physical Geology," Bull. Phil. Soc. Wash., 
XI (1889), S3- 

'Bull. Phil. Soc. Wash., XIII (1895), 31-75. 


degree of isostasy, there is but little such adjustment within areas 
200 to 300 km. in diameter, or of limited differential relief. Indi- 
vidual mountains and mountain ranges may stand by virtue of the 
rigidity of the crust. Even under the level plains equally great 
loads are permanently borne, loads produced by widespread irregu- 
larities of density not in accord with the topography above. 
Isostasy, then, is nearly perfect, or is very imperfect, or even non- 
existent, according to the size and relief of the area considered. 

The third division, the mode of maintenance of isostasy and its 
bearings on problems of the crust, remains to be considered. This 
condition of isostatic equilibrium exists at present in spite of the 
leveling surface actions and compressive crustal movements of all 
past geologic time. There must be, consequently, some internal 
mode of restoring more or less perfectly an isostatic condition, 
either by frequent small movements, or by more infrequent and 
larger ones. 

Erosion and sedimentation result in a lateral transfer of matter, 
and to maintain isostasy there must be some lateral counter- 
movement in the earth below, but in regard to how or where or 
when this is done, and as to what are its effects, there has been no 
unanimity of opinion, nor convincing demonstration. 

In considering the problems of crustal dynamics some authors 
have regarded earth shrinkage and consequent tangentially com- 
pressive forces as controlling the nature of diastrophism, including 
movements of both orogenic and epeirogenic character; others, the 
advocates of extreme isostasy, have thought to see even in folding 
only the secondary effects of movements maintaining isostatic 
equilibrium. The first point of view emphasizes the strength and 
elasticity of the crust, with long-deferred periodic discharge of 
stress. The second point of view calls for an interpretation based 
on the weakness and plasticity of the crust, with resulting nearly 
continuous small movements restoring the delicate vertical balance 
destroyed by gradational actions. To what degree are the two 
points of view compatible and within what limits is each dominant ? 
The problem of this chapter involves, therefore, not only the mode 
but the limits and effects of the movements which more or less 
completely maintain or restore isostasy. 


The method of attack is largely one of exclusion. By showing 
what hypotheses cannot apply, the way is prepared for conclusions 
in better accord with the fields of fact and theory. 

The results show that conditions of isostatic equilibrium cause 
the light and high segments to press heavily against the adjacent 
lower and heavier ones, most heavily above. The tendency is 
consequently for the high areas to spread with a glacier-like flow 
over the low areas. This tendency, however, is effectively resisted 
by the strength of the crust. Upon the disturbance of equilibrium 
by erosion and deposition there are two kinds of stresses produced 
which tend to restore equilibrium. The first is a tendency of the 
heavy column to underthrust the lighter, but it could never produce 
compression and folding at the surface. This force would be most 
effective under the hypothesis of great crustal weakness, so that 
the vertical stresses could be transmitted in a horizontal direction 
within the lithosphere as in a fluid. Even in that case, however, 
it would not be the dominating force. The actual isostatic move- 
ments consist of a rising of the eroded areas, a sinking of those 
which are loaded. This involves shear or flexure around their 
boundaries. The columns must be large enough so that the excess 
or deficiency of mass can become effective in producing deformation. 
When the accumulating vertical stresses have overcome the strength 
of the crust, the excess pressure from the heavy area is transmitted 
to the zone below the level of compensation. This deep zone is in 
turn the hydraulic agent which converts the gravity of the excess 
of matter in the heavy column into a force acting upward against 
the lighter column and thus deforms the crust of the eroded area. 
By this means even the continental interiors are kept in isostatic 
equilibrium with the distant ocean basins. This implies a great 
depth and thickness to the zone of plastic flow. Although it must 
be plastic under moderate permanent stresses, this does not imply 
by any means a necessarily fluid condition, and fluidity is disproved 
by other lines of evidence. 

The zone of compensation, being competent to sustain the 
stresses imposed by the topography and its isostatic compensation, 
must obey the laws pertaining to the elasticity of the solid state 
and is to be regarded therefore as of the nature of rock. Conse- 
quently there may be extended to all of it the name of the litho- 


sphere, even though it includes from time to time molten bodies, 
the constituents of the pyrosphere. 

The theory of isostasy shows that below the lithosphere there 
exists in contradistinction a thick earth-shell marked by a capacity 
to yield readily to long-enduring strains of limited magnitude. 
But if such a zone exists it must exercise a fundamental control 
in terrestrial mechanics, in deformations of both vertical and tan- 
gential nature. It is a real zone between the lithosphere above 
and the centrosphere below, both of which possess the strength to 
bear, without yielding, large and long-enduring strains. Its reality 
is not lessened because it blends on the limits into these neighbor- 
ing spheres, nor because its limits will vary to some degree with the 
nature of the stresses brought upon it and to a large degree by 
the awakening and ascent of regional igneous activity. To give 
proper emphasis and avoid the repetition of descriptive clauses it 
needs a distinctive name. It may be the generating zone of the 
pyrosphere; it may be a sphere of unstable state, but this to a 
larger extent is hypothesis and the reason for choosing a name 
rests upon the definite part it seems to play in crustal dynamics. 
Its comparative weakness is in that connection its distinctive 
feature. It may then be called the sphere of weakness — the 
asthenosphere, and its position among the successive shells which 
make up the body of the earth is as follows: 

The atmosphere 

' Including the biosphere 
The hydrosphere 

The lithosphere 

' Including the pyrosphere 
The asthenosphere 

The centrosphere, or barysphere 

Each has played its fundamental part in the development of 


Stresses under conditions of isostatic equilibrium. — The conti- 
nental platforms slope down into the ocean basins at grades which 
range mostly from one in ten to one in thirty. Some of the great 


foredeeps show both the greatest depths of water and the steepest 
descents. The Chilean coast, for instance, at lat. 25° S., slopes 
from the Andes to a depth of 7,500 meters with a submarine grade 
of one in eight. Under the hypothesis of nearly perfect isostasy, 
which will be favored in this discussion, this would be taken to 
show the contiguity of areas in the crust of markedly unlike density. 

Let the slope between such areas be regarded as a thick parti- 
tion between two columns, each in isostatic equilibrium. These 
rest then upon the substratum below the zone of compensation 
with the same pressure and stand vertically in equilibrium. 

In so far as the rock within the crust is subjected to mere cubic 
compression, equal in all directions and increasing with depth, 
there is no distortional force. In so far, however, as side pressures 
in one column are not balanced by equal side pressures from the 
adjacent columns, there is a stress-difference which does produce 
a distortional strain. If the stress-difference exceeds the elastic 
limit a permanent deformation results which reduces the stress 
and eases the strain. It is the plan of this paper to discuss the 
nature of the stress-differences on the partition separating two 
contiguous columns of the crust, of markedly unlike density; 
first, when these are in isostatic equilibrium, and second, when 
not in such equilibrium. Fig. 13 is drawn to show graphically 
these relations. 

The land-column of the crust is marked M; the submarine 
column is N; is the earth-shell below the zone of isostatic com- 
pensation; P is the column of sea-water. The vertical partition 
between the unlike columns stops in reality, according to the hy- 
pothesis, at the bottom of the columns. It is here extended down 
through the earth-shell 0-0 in order to discuss the deformation 
which would take place in the latter shell. M and N represent 
what is here called the lithosphere; 0-0 the zone which it is 
proposed to call the asthenosphere. 

In case A, isostatic equilibrium is assumed and the pressures 
of the two lithospheric columns are equal upon the asthenosphere. 
But, assuming for the moment that the vertical pressures are freely 
transmitted as lateral pressures, it is seen that a marked horizontal 
unbalanced pressure is produced by the land-column against the 



sea-column, as represented by the horizontal lines of the stress 
diagram. The top of the land-column is balanced only against the 
negligible weight of the atmosphere and the lateral stress gradient 
is there highest. The next portion below is balanced against the 



-^-^ p 

3 N 



Stress scale 

assumed as 
Densities 1 

M?70:N2.77 1 


- level 

JT*..,.. P„ 











3 N 



Fig. 13. — Diagram illustrating pressure-relations of the crust for marginal 
portions of the continental shelf and oceanic basin, interpreted as balanced by uni- 
formly distributed isostatic compensation. Stress-differences are shown by cross- 
lined diagrams, the pressures being regarded as transmitted hydrostatically. The 
actual lateral stress-differences, for stresses within the elastic limit, are about one- 
fourth of the hydrostatic pressures here shown. 

A. Columns in isostatic equilibrium. 

B. Relations after base-leveling. 

C. Relations after re-establishment of isostatic equilibrium. 


sea-water and the stress gradient becomes less high. The maximum 
thrust occurs at the bottom of the ocean and is from the land toward 
the sea. Below this level the density of the sea-column is greater 
than that of the land-column. This, with increasing depth, 
gradually balances the excess pressure, and at the base of the litho- 
sphere both the lateral and vertical pressures of both columns by 
hypothesis are equal. 

In this diagram the pressures of the columns are imagined to 
act hydrostatically, but, in reality, for stresses within the elastic 
limit, this would not be so. Further, in so far as the partition is 
much wider than the difference in elevation of the columns, it has 
a gentle surface slope and will tend to give the upper part of the 
land-column competence to hold itself in by its own strength and 
that of the partition. The approximate ratio which the actual 
lateral pressure-differences on the two sides of the partition hold to 
the assumed hydrostatic pressures may be perceived from the 
results of a recent work by Love entitled Some Problems of Geo- 
dynamics. 1 In chaps, ii and iii he considers the problems of the 
isostatic support of continents and mountains. As a basis for the 
analytic treatment he assumes, first, the existence of complete com- 
pensation within a depth of one-fiftieth of the earth's radius, 
127 km.; second, that at this depth all stress-differences disappear, 
the pressures below being of the nature of hydrostatic pressures, 
the only kind which could occur if a fluid layer existed at and below 
the depth of 127 km.; third, it is known that the heterogeneities of 
mass in the lithosphere only slightly modify the form of the geoid, 
and it is accordingly assumed that there is no such effect. Love 
thus treats of the limiting case of a crust exhibiting perfect isos- 
tasy, its surface relief not modifying the form of the geoid given by 
the ocean surface, and resting with its base upon a fluid zone. As 
such, his solution is of great value, but he states: "It must, how- 
ever, be understood that the special form (of the hypothesis of 
isostasy) is introduced for the sake of analytical simplicity rather 
than physical appropriateness." 2 

The artificiality of the assumption of the existence of no stress- 
differences below the zone of compensation is shown by the law of 

1 Cambridge University Press, 1911. ' Op tit., p. 7. 


density distribution which results. With only these three limiting 
assumptions, the number of unknown quantities remains larger 
than the number of equations, and the results are, strictly speaking, 
indeterminate; but by making various reasonable further assump- 
tions definite solutions in accordance with these may be obtained. 
The elimination of stress-differences at the base of the lithosphere, 
taken as equivalent here to the zone of compensation, requires, 
however, that there shall be a peculiar relation of densities. To 
compensate an elevation it must be offset by matter below of less 
density than the mean for that depth, but in order to quench the 
stress-differences at the base of the lithosphere there must be 
between the light matter and this base a layer of more than mean 
density for that depth. Thus the light layer must perform a two- 
fold function, compensating not only the elevation above but the 
heavy layer below. For depressions in the crust there must be a 
reverse arrangement, matter of more than mean density existing 
immediately below the surface. But above the base of the litho- 
sphere there must be a layer of less than mean density for that 
depth. The artificialities of this scheme would be sufficient to 
form a disproof of the initial assumption which determined it, but 
it also seems to be directly disproved by the evidence brought 
forward in the earlier parts of the present article. Nevertheless, 
the exact mathematical solution of this difficult problem is of great 
value as giving the results of the assumptions of extreme isostasy. 

For the largest inequality of the crust, regarded as a zonal 
harmonic of the first order, that represented by the land and water 
hemispheres, Love shows that the lateral stress-differences under 
this hypothesis of isostasy reach a maximum at a depth equal to 
one- third of the zone of compensation and are equal to only o . 006 
of the weight of a column of rock of height equal to the maximum 
height of the inequality. For harmonics of the second and third 
orders, representing the continents, the fractions are 0.0134 and 
0.0208. These results, Love states, are extremely favorable to the 
hypothesis of isostasy, since the inequalities could be supported by 
any reasonably strong material. 

There are two criticisms, however, to be noted while citing this 
conclusion. First, it is known that the crust is vastly stronger than 


these requirements, so that such a perfected isostatic arrangement 
is not demanded on the score of crustal weakness. Second, the 
harmonic curves giving these figures are of a gently sweeping 
character; whereas, the actual continents are in many places high 
on their margins, and from these margin's they slope with compara- 
tive steepness to the mean depth of the ocean floors. The stresses 
set up beneath the continental margins ar*» accordingly a closer 
approximation to those imposed by lofty mountain ranges. Assume 
that compensations of the continental margins are perfect and the 
problem becomes that which Love takes up in the following chap- 
ter, namely, the isostatic support of mountains, except that we deal 
with only one great slope, whereas the theory calls for a succession 
of mountains and valleys. 

It is shown that for such a compensated series, postulating the 
distribution of densities previously discussed, the greatest stress- 
difference exists at the mean surface, beneath the crests, and 
reaches a value equal to half the weight of a column of rock equal 
to half the height of the crests above the valley bottoms. From 
this maximum the stress-difference decreases to zero at the base of 
the zone of compensation. The solution by G. H. Darwin for 
uncompensated mountains and valleys gave a maximum stress- 
difference equal to 74 per cent of half the height, this maximum 
occurring at a depth equal to about one-sixth the distance between 
mountain crests. Even with perfect isostatic compensation, dis- 
tributed after the fashion assumed by Love, the stress-differences 
for mountains and valleys are seen consequently to be two-thirds 
in value of those produced by an uncompensated relief, and are 
approximately one-fourth of the hydrostatic pressures. This frac- 
tion, one-fourth, happens also to be the same as Poisson's ratio, 
the ratio of the lateral expansion to the vertical shortening of a 
free rock column under vertical stress. 

Now the distribution of density has been found to be more or 
less irregular, and there is no evidence of such a reversing layer at 
the base as Love has postulated. Stress-differences will conse- 
quently extend below the isostatic compensation. If, however, the 
latter is not uniformly distributed, but is concentrated somewhat 
in the outer half of the lithosphere, the stress-differences will become 


small at and below the base of the lithosphere. On account of the 
incompleteness of local compensation, the irregularities and uncer- 
tainties of the actual facts of nature, the Gordian knot of a solution 
may be cut by simply assuming for present purposes the form 
of diagram given by hydrostatic pressures due to a compensation 
uniformly distributed. The approximate stress-differences will be 
given by taking one-fourth of the values given by the hydrostatic 
pressures. This transfers the problem from the difficult field of 
zonal harmonics to the simple one of hydrostatics, and perhaps does 
not introduce errors greater than those involved in the differences 
between nature and the postulates which form the foundation of 
the solution by zonal harmonics. This hydrostatic diagram is 
shown accordingly in Fig. 13. 

It is held by the advocates of extreme isostasy, however, that 
for long-continued stresses the crust is very weak; in other words, 
the elastic limit is low, and slow plastic deformation readily 
occurs which tends to dissipate the stress-differences and re- 
establish isostatic equilibrium. To the extent to which this 
is true, the real diagram of lateral stresses would approach the 
hydrostatic diagram here given and measure the forces producing 
plastic flow. 

It has remained, however, for the opponents of the hypothesis 
of local and nearly perfect isostasy to point out, what is here 
illustrated graphically, that the extreme theory requires a belief 
in vertical weakness but lateral strength. If it were not for lateral 
strength the land-column would crowd against the sea-column, 
more at the top than at the bottom. Flowing out with a glacier- 
like motion over the upper part of the sea-column, the land-column 
would settle at the top and become shorter. This in turn would 
bring about a vertical elevatory pressure against its bottom, the 
column would rise, lateral creep would continue with equal pace, 
and the end result would be a density stratification in which the 
continental crust would come to overlie the oceanic crust. The 
limit of such an action would be given by the decreasing surface 
gradient, this finally becoming so gentle as to stop the glacier-like 
flow. The lack of such an effect implies of course that the lateral 
stresses of the outer part of the lithosphere lie within the elastic 


limit. Therefore they may be regarded as having not more than 
a quarter of the value shown in Fig. 13A. 

The suggestion of the existence of opposing modifying factors is 
to be found in conclusions from the previous parts of this investi- 
gation — that compensation may be in many places concentrated 
somewhat in the outer half of the zone as here shown and in other 
places fade out through a notable distance below. These two 
variations in the distribution of compensation would modify the 
stress diagram in opposite directions. 

Modifications of stresses produced by base-leveling. — Consider next 
the case of complete erosion to sea-level, as shown in Fig. 13B. 
The rock from the land-column has been deposited as sediment 
over the sea-column. As the columns are supposed to act as 
units the sediment is shown as spread uniformly. The lateral 
stress diagram beneath the bottom of the sediment shows a rate 
of decrease the same as in case A, but the value of the hydrostatic 
stress at any depth is diminished by the sum of the depths of 
erosion and deposition. The lateral stress now changes in sign 
at a point 5 and at this depth is a line of no lateral stress. Above 
this depth the continental segment tends still to spread over the 
ocean, but less effectively than before; below this depth the oceanic 
segment now thrusts against the continental crust. 

If the ocean water be eliminated from the diagram and base- 
leveling should bring both columns to a uniform surface, then the 
neutral depth S advances to the surface and the lateral stress 
diagram in B is just the reverse in value to A. In that case there 
is no lateral thrust at the surface, but at all depths below there is 
an excess pressure against the continent reaching a maximum 
at the bottom of the lithosphere. This extreme case cannot apply 
to the ocean except for that limited width over which is built out 
a continental shelf. To the degree to which the weight of this 
shelf is supported by the ocean crust beyond, the column beneath 
the shelf would not operate with its full pressures against the land. 
The case would apply better to the complete erosion of level- 
topped plateaus situated within a continent. 

For the lateral pressure within the lithosphere to become effect- 
ive in a landward undertow would require a lesser rigidity of the 


crust at the bottom than at the top. Such a lesser rigidity may be 
granted, but it is seen then that the landward undertow would be 
greatest at the bottom and could not advance above a depth 
indicated on the diagrams by T. At this point the stress is of the 
opposite sign but of the same value as for the state of isostatic 
balance in case A. If seaward flow did not take place at this level 
in the first case, landward flow could not take place in the second. 

For the extreme case where isostasy is completely destroyed by 
surface leveling, no water body remaining, T will rise upward to a 
depth equal to one-half the depth of the zone of compensation. 
If the surface of complete compensation be 76 miles deep, this 
gives a minimum depth of 36 miles. For the undertow to reach 
this height implies, however, not only the limiting case of complete 
destruction of isostasy, but a crust only one-half as rigid at depth 
T as at the surface and a previous state of expansive surface 
stress as great as the outer crust could bear. On the other hand, 
if tangential pressures due to centrospheric shrinkage should 
co-operate with the stresses tending to restore isostatic equilibrium, 
underthrust would become more effective below, but overthrust 
would also become effective above. 

The disappearance of isostatic compensation at a certain level 
means the disappearance of notable heterogeneity in the earth- 
shell below, as argued in Part V (pp. 446-48). One of the possible 
suppositions to explain this is to suppose that this shell is weaker 
than the crust above and therefore the lateral thrust due to an 
assumed initial heterogeneity would cause a lateral flow, a density 
stratification, and a resulting disappearance of the postulated 
heterogeneity. This supposition of a weaker zone finds support 
in other lines of evidence. Therefore, although some lateral flow 
at the base of the lithosphere may occur during the restoration of 
isostatic equilibrium, it is to be expected that the bulk of such 
flow will be below, for there the substance is more plastic and the 
lateral stress is throughout at a maximum. 

Let attention be given next to the vertical as contrasted to the 
lateral unbalancing brought in by the destruction of isostatic 
equilibrium. The land-column becomes lighter, the sea-column 
heavier, by amounts which are shown in the vertically lined stress 


diagrams at the base of the lithosphere in case B. Supposing that 
vertical readjustment of the columns is prevented for a time by the 
strength of the crust, the vertical stresses will be taken up by a 
vertical shearing strain along the partition between the two 
columns. This shear is equal in amount to the difference in total 
weights of the columns. Let the shear per unit area be called 5. 
It acts over a surface taken as 122 km. high. Let this height be 
called h. The weight of the columns will vary with their breadth 
in the plane of the drawing. If the breadth of each be taken as b 
and the weights per unit area as M and N (N including rock, sedi- 
ment, and sea- water), then for a cross-section of unit thickness 
the total difference in weight is (N—M)b and the total shear 
is this same amount, provided that the columns are not sus- 
tained in part by other boundaries. But the total shear is also sh. 


s=(N-M)- k 

For narrow columns b is small, giving to s a small value and con- 
sequently one within the elastic limit. Let b become broad and 5 
will then become large and exceed the elastic limit. The lateral 
pressures, on the contrary, are less dependent upon the breadth, 
and, if the problem were regarded as one of hydrostatic pressures, 
would be wholly independent of breadth. The formula shows that 
the broader the columns, the more readily they will readjust by 
vertical shear between the columns. Now unless failure by vertical 
shear took place between the upper part of the columns the heavy 
column would be held up, the light column would be held down, 
except for the partial effect of sagging in case the columns were 
very broad. The lateral landward pressure at the base could 
therefore not become effective. The loaded portion of the crust 
must fail first by shear or flexure of its upper portion. Whatever 
be the distribution of strength it would appear then that the primary 
yielding is the vertical one and the landward force of undertow 
can become only secondarily effective. 

The hypothesis of local and nearly complete isostasy requires 
that the elastic limit for vertical shear should be very low in order 


that narrow columns should be able to rise or sink. This may be 
illustrated by the following example: 

Suppose a region 50 km. in radius possesses a mean departure 
from isostatic equilibrium equal to 76 m. of rock (250 ft.) and that 
the surrounding regions are out of adjustment by the same amount 
but in the reverse direction. This is the maximum area for regional 
isostasy which in Hayford's opinion is to be expected, and 250 ft. 
is the mean departure from isostasy as given by him in his Minne- 
apolis address. But in this example the adjacent regions are each 
assumed to be out of adjustment in opposite directions by this 
amount and, therefore, the differential load is twice this or 500 ft. 
of rock. The case is one which he would regard consequently as 
rather extreme. Now a cylinder 100 km. in diameter and 122 km. 
deep could not fail through its bending moment, as in the flexing 
of a beam. It would have to fail as in punching a rivet hole 
through a metal plate, in other words, by circumferential shear. 
The shearing stress per unit area is obtained by dividing the total 
load by the total shearing surface. With the data taken as above 
this gives 5=8.4 kg. per sq. cm. or 120 lbs. per sq. in. But strong 
rock at the surface can readily carry a shearing stress of from 700 
to 1,000 kg. per sq. cm. (10,000 to 14,000 lbs. per sq. in.). Isostatic 
perfection to this degree would therefore require the zone of com- 
pensation as a whole to be only about one-hundredth as strong under 
permanent stress as is solid rock at the surface. This calculation 
alone would tend to show that the loads and areas by which the 
crust departs from isostatic equilibrium have been much under- 
estimated by the advocates of extreme isostasy. 

It should be noted, however, that, following the lines of his 
rejoinder to Lewis, Hayford would answer that he regarded the 
landward isostatic flow as taking place within the zone of isostatic 
compensation and the vertical shear as operating, consequently, 
through a depth far less than the thickness of the entire zone of 
compensation. There are, however, a number of inconsistencies 
in this argument, some of which have already been made evident. 
Others will appear as a result of the later discussion of this chapter. 
But it may be noted that even granting this contention — that only 
the outer third of the zone of compensation was involved — the 


unit shearing stress would be multiplied only by two or three and 
would still imply a weakness in this part of the crust to resist 
long-enduring shear or bending stresses, its capacity being only 
3 or 5 per cent at most as great as is found to exist in surface rocks 
for stresses of human duration. 

Relief of stress accompanying restoration of isostasy. — It is seen 
from the preceding analysis that the movement of the unbalanced 
columns toward a new state of equilibrium will be partly by vertical 
shear in the neutral ground between them, but, where the areas 
are large in comparison with the thickness of the zone of compensa- 
tion, the easiest mode of yielding may be by flexure, showing at the 
surface as crustal warping. Both modes of yielding serve to trans- 
mit the excess vertical stresses of the heavy and sinking column 
into the asthenosphere. If the latter be indeed a shell of weakness 
it will transmit these pressures more or less hydrostatically. The 
vertical pressure- differences will act within it as lateral pressures 
making for flow toward the lighter column. It is shown in Fig. 13B 
that the maximum horizontal stress in so far as it approaches 
a hydrostatic distribution acts throughout the whole depth of this 
zone, so that it not only is weaker than the crust above, but is 
subjected to maximum stress over a greater area. It will yield 
by flowage therefore either if of small depth and very plastic, or of 
great depth but more rigid. If the columns are adjacent and nar- 
row as compared to the thickness of the shell of weakness, then the 
principles of plastic flow would require that the flow be chiefly in 
the upper part of this shell. If, however, the columns are of con- 
siderable breadth compared to the thickness of the asthenosphere, 
and especially if at a distance from each other, then the principle of 
least work would determine that the middle strata of this shell 
should flow the farthest and the whole would to some degree 
participate. If an imaginary partition were extended downward 
through this shell as shown in A and B of Fig. 13 this partition 
would be found warped after the movement as shown in C of the 
same figure. 

It was seen in an earlier part of this discussion that, even sup- 
posing deformation became effective by means of the lateral 
stresses within the lithosphere and without the existence of a zone 


of weakness below, still only the basal part below the point T 
would be competent to give a landward movement during the 
restoration of isostatic equilibrium. But now it is seen that 
in the asthenosphere the lateral pressures are transmitted 
with greater amount, from a greater distance, and with a greater 
cross-section. The zone is one without notable isostatic compen- 
sation within it and is presumably more plastic than the 
basal part of the lithosphere. Therefore there is good reason 
to believe that the subcrustal undertow is restricted to the 

The forces actually needed to produce flowage would be in 
reality but a fraction of those indicated in Fig. 13B as existing in 
the asthenosphere. The reason is that the greater part of the 
vertical forces is consumed in producing flexure and shear in the 
lithosphere. Only a residuum is needed to produce a slow plastic 
flow in the shell below. For that reason broken lines are used in 
that part of the stress diagram. The energy consumed within 
the lithosphere by its deformation will be nearly independent of 
the breadth of the columns; it will actually tend to become some- 
what less with breadth because flexure on large radii will be favored. 
The energy consumed in the asthenosphere will, on the other hand, 
increase with the breadth of the columns, but will be spread over 
a greater area. The temperature effect due to the absorption of 
energy would appear to be a minor factor, for it cannot exceed that 
energy which is supplied by the average vertical stress-difference 
multiplied by the vertical distance moved. The average vertical 
stress-difference will be the mean between that at the beginning 
of movement and that residual stress remaining after the movement 
is completed. 

In determining the scale of the diagrams of Fig. 13 the following 
data were chosen. The land-column was taken in A as having a 
surface elevation of 1,000 m. and a density of 2.70; the sea as 
3,000 m. deep, and the rock below as possessing a density of 2. 77. 
The sea- water has a density of 1.03. These relations give an 
isostatic balance at a depth of 122 km. In B, erosion of the land 
to sea-level is supposed to have taken place and the sediment 
spread with same unit weight over the sea-column that it had as 


rock upon the land. These relations give a depth of 54 km. to 
S and 88 km. to T. 

It should be repeated, however, in closing this topic, that the 
solutions here given are approximate only and assume that iso- 
static compensation results in lateral stress-differences which show 
the same distribution of forces as a diagram of hydrostatic pressures, 
differing only in magnitude. The writer is inclined to think that 
the actual facts of nature call in most cases for some depression in 
depth of the critical points beyond those here shown. Especially 
is there likely to be under the margins of a continent in isostatic 
equilibrium some permanent lateral stress-difference within the 
asthenosphere, due to the compensation above and tending toward 
a landward undertow. Upon the unbalancing due to erosion and 
sedimentation this would cause the lateral stress-differences within 
the asthenosphere to rise more quickly to the low elastic limit and 
permit more readily than would otherwise be the case a regional 
readjustment toward isostasy. 


Present status of the problem. — The causes of vertical movements 
Dutton 1 made twofold. He clearly distinguished on the one hand 
between those internal forces leading to expansion or contraction, 
which tend, by producing changes in density, to create isostatically 
a new surface relief, and, on the other hand, those isostatic re- 
adjustments following erosion and sedimentation, readjustments 
which tend not to make a new, but to restore the older, relief. 
Folding he regarded as unrelated to the former, as a result of the 
latter. He had shown earlier (in fact, he had the honor of being 
the first to show) that the time-sanctioned hypothesis of cooling 
as a cause of crustal shrinkage and consequent mountain-making 
was inadequate to account for either the distribution or amount of 
folding. 2 From this he was led to regard folding as due, not to any 
kind of contraction, but as a compressive movement of one section 

1 "On Some of the Greater Problems of Physical Geology," Bull. Phil. Soc. Wash., 
XI (1889), 51-64. 

* C. E. Dutton, "A Criticism upon the Contractional Hypothesis," Am. Jour. 
Sci., VIII (1874), "3-23- 


of the crust against another, presumably offset by tension in some 
other region. Dutton's argument is that the crust beneath the 
plateau is unloaded by erosion, that the crust beneath the basin is 
loaded by sedimentation. An isostatic movement, rejuvenating 
the relief , must, by causing the overloaded basin to settle, produce 
a squeezing-out of matter beneath the sinking area, and a crowding- 
in of matter beneath the rising area. The surfkial movement of 
sediment is from the high area toward the low. The deep-seated 
movement is from the low toward the high. Thus the cycle 
becomes completed and the mass of matter above the level of com- 
plete compensation remains the same in each column. The seaward 
movement of the sediment, as a frictional resistance against the 
river bottoms, produces only an insignificant drag, but the return 
subterranean movements by viscous or solid flowage must produce 
a pronounced drag upon the crust in the direction of the rising 
region. Dutton's reasoning is clear, but the effectiveness of the 
action rests upon several assumptions. First, it omits the influence 
of the surface relief and the degree to which that tends to a lateral 
spreading movement from the high toward the low regions. Sec- 
ondly, it postulates a low rigidity to the crust, as he in fact notes. 
Thirdly, it involves the conception of a strong undertow fairly 
near the surface in order that the crust above may be too weak to 
resist the viscous drag. As there were little quantitative data 
available at the time when Dutton formulated this corollary of his 
theory of isostasy he could not have tested the validity of these 
assumptions, but raised the problem for those who should come 
after him. 

This theory of folding took a somewhat different form in the 
mind of Willis, as expressed in the concluding chapter of his 
Research in China. 1 This work in many ways is of the very first 
importance and gives a comprehensive view of the geological history 
of the whole continent of Asia. As to the nature of the movements, 
he finds that the continent of Asia may be resolved into positive 
and negative elements, the former areas tending to stand high, the 
latter tending to stand low. These tendencies are latent during 
comparatively long periods of quiet and resultant peneplanation, 

1 Vol. II (1907), Carnegie Institution of Washington. 


but become operative during epochs of diastrophism. The com- 
pressive movements, on the other hand, have pressed and welded 
the positive elements together, the axial directions of folding 
representing the compression of the negative zones lying between. 

The cause of the diastrophism Willis ascribes to differences in 
specific gravity, restricted, according to Hayford's determination, 
to the outer hundred miles of the earth's body; the vertical move- 
ments being chiefly due to isostatic readjustment between the 
several continental elements, the compressive movements being 
due to the tendency of the heavier oceanic segments of the earth 
to spread and underthrust the outer portions of the whole conti- 
nental mass. This theory of the cause of lateral compression was 
discussed by the present writer in a review of Willis' work, 1 and 
the objections stated against it there are in part the same as will 
be elaborated farther on in the present article. 

Hayford took up the same subject in his address, delivered at 
Minneapolis on December 29, 1910, as retiring vice-president of 
Section D (Mechanical Science and Engineering) of the American 
Association for the Advancement of Science, the title of his paper 
being "The Relations of Isostasy to Geodesy, Geophysics, and 
Geology." 2 This is a paper of broad scope intended to show how 
vertical movements not in apparent accord with isostasy and also 
movements of folding may be explained as secondary results of 
isostatic adjustment and really in harmony with the hypothesis 
of nearly continuous movement in a crust of low rigidity and of 
almost complete isostasy. This part of his theory is essentially 
the same as Dutton's but is elaborated in greater detail. 

Harmon Lewis called attention to the defects in this theory of 
deformation, 3 but Hayford made a rejoinder, positive and sweeping 
in its style, to this and other lines of criticism by Lewis. 4 

The names of Dutton, Willis, and Hayford deservedly carry 
much weight and must be accepted at their face value by geologists 

1 Science, N.S., XXIX (1909), 257-60. 
3 Published in Science, N.S., XXXIII (1911), 199-208. 
J "The Theory of Isostasy," Jour. Geol., XIX (1911), 620-23. 
■•John F. Hayford, "Isostasy, a Rejoinder to the Article by Harmon Lewis," 
Jour. Geol., XX (1912), 562-78. 


who have not themselves made a critical study of the problems of 
isostasy. The arguments which the writer advanced in 1909 
against this hypothesis were published under a title which appar- 
ently did not call attention to them. The style of Hayford's reply 
to Lewis is crushing and conveys the impression that Lewis has been 
completely refuted. It is because of these reasons that the sub- 
ject calls here for fuller development. 

In his Minneapolis address Hayford outlines a theory of the 
principles of diastrophism which turns upon his conclusion that 
isostasy is so nearly complete that areas of even limited size average 
only 250 feet from the level of isostatic equilibrium. He assumes 
chemical and physical changes to be induced in the crust by 
the changing load due to erosion and sedimentation. These he 
thinks are superimposed upon the effects of nearly continuous 
vertical movements of isostatic readjustment. The vertical move- 
ments in turn produce a lateral undertow which is given as a cause 
of localized heating and folding. Apparently this is regarded as 
a complete mechanism of deformation since the author raises the 

Is it at all certain that under the influence of such actions the geological 
record at the earth's surface at the end of fifty to one hundred million years 
would be appreciably less complicated than the geologic record which is actually 
before us ? I think that it would be fully as complicated as the actual record. 1 

This theory of folding as the result of subcrustal undertow is 
illustrated by means of two diagrams. In Fig. 1 , the zone of viscous 
flow from the sinking toward the rising area is placed in the lower 
quarter of the zone of isostatic compensation. In Fig. 2 it is shown 
in the middle of that zone, dying out both above and below. 
Apparently then, as shown by these two different conceptions, the 
author cited was guided by no definite theory, based upon the 
mechanics of materials, as to the factors which would determine 
the depth of this zone of undertow and its relations to the zone of 

Harmon Lewis in his paper on the "Theory of Isostasy" has 
discussed various aspects of the isostatic theory as developed by 

1 Op. tit., p. 206. 


Hayford, and among them this question. Regarding the possi- 
bility of folding by means of isostatic undertow, Lewis concludes: 

Now, according to the theory of isostasy, compensation would be essen- 
tially complete, and if compensation is complete the depth of compensation 
as determined by Hayford's geodetic work would be as great as 60 miles. 
Hence, the undertow postulated by isostasy would exist chiefly below 60 miles. 
It is decidedly questionable that an undertow even much nearer to the surface 
than 60 miles would cause the observed folding in the upper few miles of the 
crust. 1 

In regard to this criticism by Lewis concerning the cause of 
folding, Hayford states in reply: 

On pp. 621-22 Mr. Lewis sets forth the argument that there is much 
geological evidence of horizontal movements in the outside portions of the 
earth, especially in the form of folding, that the controlling movements of 
isostasy are assumed to be vertical and hence cannot account for folding, and 
that the horizontal movement or undertow concerned in isostatic readjustment 
must be below the depth of compensation and hence so far below the surface 
as to be very ineffective in producing folding. 

There are two fatal defects in this argument as applied to controverting 
anything that Hayford believes or has written. 

First, Hayford has already indicated clearly his belief that the undertow 
concerned in isostatic readjustment is above, not below, the depth of compensa- 
tion. In both the figures published in his Minneapolis address the undertow 
is clearly indicated as being above the depth of compensation and it is also 
so indicated in the corresponding text. As Hayford puts the undertow com- 
paratively near the surface, where it is conceded that it would be effective in 
producing folding, the existence of extensive folding is a confirmation, not a 
contradiction, of his theory of the manner in which isostatic readjustment 
takes place. It is certainly not fair to hold Hayford responsible, either directly 
or by inference, for any theory which someone else may believe which involves 
an undertow situated entirely below the depth of compensation. Mr. Lewis 
apparently believes such a theory. 

Second, the movements which produce isostatic readjustment are neces- 
sarily horizontal, not vertical. If two adjacent columns of the same horizontal 
cross-section extending from the surface to the depth of compensation have 
different masses the readjustment to perfect compensation must involve a 
transfer of mass out of one column, or into the other, or from one to the other. 
In any case the transfer must be a horizontal movement. Hayford has already 
shown in print more than once that he understands that vertical movement 
alone does not produce isostatic readjustment. Moreover, a careful reading 

x Op. cit., p. 622. 


of his Minneapolis address will certainly show that he believes that the total 
amount of material moved horizontally during isostatic readjustment, and 
especially the total number of ton-miles of such movement, is vastly in excess 
of the corresponding quantities concerned in the vertical components of the 
movement which takes place. Hence the folding and other abundant evidence 
of past horizontal movements observed by geologists confirm Hayford's 
hypothesis as to the manner in which isostatic readjustment takes place, 
instead of conflicting with it as Mr. Lewis' article would lead one to think.' 

The present writer, however, believes with Mr. Lewis in the 
theory that an undertow must be essentially below the zone of 
compensation and is incapable of producing surficial folding. The 
reasons have been given in part in the consideration of the stress- 
relations, as they would exist under the hypothesis of extreme 
isostasy. But there are other reasons why the subject should be 
discussed in further detail. One reason is that, if Lewis is right 
on this point and Hayford wrong, it is desirable that this should 
be made clear, in justice to Mr. Lewis as well as to the subject. 
The other reason is that here in reaching a conclusion we can 
advantageously pursue a method of exclusion. By showing that 
isostatic undertow cannot take place within the zone of compen- 
sation, for various reasons besides those discussed in the stress 
diagrams, we reach the conclusion that it must take place in a 
level below that zone. Furthermore, by noting the conditions 
which would hinder lateral flowage we may arrive at a conclusion 
as to those which must exist to greater or less degree in order to 
permit it. 

Objections against undertow in the zone of compensation. — The 
pressures which occur during a state of isostasy and after the 
destruction of that condition have been discussed. It was seen 
that the pressures making for the undertow necessary to restore 
isostasy were greatest at the bottom, but, more especially, below 
the bottom of the zone of compensation. The possibility remains 
to be considered, however, that perhaps the distribution of the 
rigidity of the crust more than offsets the distribution of pressures. 
Suppose the middle of the zone of compensation should be very 
weak and the crust at and below the bottom be very strong. Then, 

1 Op. tit., pp. 573, 574- 


if the restoration of isostasy was deferred until assisted by strong 
tangential pressures due to centrospheric shrinkage, it might be 
held that isostatic undertow could take place within the zone of 
compensation and between S and T of Fig. 13B. If, furthermore, 
compensation should be not uniformly distributed but taken as 
largely concentrated in the upper part of the zone of compensation, 
which however is contrary to the Hayfordian hypothesis, then the 
forces making for undertow may correspondingly rise in the crust. 
For these reasons it is seen that the previous argument from the 
distribution of pressures is not final and that the physical conditions 
involved in lateral flowage must also be considered. 

The only positive reason which has been advanced for seeking 
to place the undertow within the zone of compensation is in order 
to utilize its viscous drag as a cause of folding. To become effective 
the drag must be strong, the crust above by contrast weak and 
therefore thin. The crumpling pressure on the surface of the 
crust cannot be transmitted directly from the sinking area, as is 
shown in Fig. 13, since the thrusting force is greatest at the bottom. 
It must be supposed to arise from the viscous drag of the undertow 
But viscosity decreases the hydrostatic head with increasing dis- 
tance from the source. Therefore, to permit a viscous flow at 
a distance from the source of pressure implies a mobility within 
that level of the crust which would make it wholly incapable of 
carrying the stresses necessary to maintain its own isostatic 
equilibrium. Therefore this level, by the very terms of the general 
conception of isostasy, would become the bottom of the zone of 

As another mechanical defect of the theory under review, it 
is to be noted that the section of undertow taken by Hayford as 
in the middle of the zone of compensation is not given as involving 
more than half of that zone. This is as if a viscous fluid were 
transmitted through a pipe in which the cross-section of pipe and 
fluid were equal. To assume that the fluid is free to escape into 
a region of less pressure at the far end and yet gives such a fric- 
tional resistance against its walls as to be able to crumple up the 
pipe is to assume that the two are of the same order of strength. 
The materials of pipe and fluid might almost be interchanged. 


In such viscous flow the tendency would be for a swelling and burst- 
ing to appear at the near end rather than a through flowage with 
a crumpling of the pipe at the far end. 

Finally, the greatest theoretical difficulty is encountered when 
it is sought to transmit matter from beneath the regions of oceanic 
marginal sedimentation to beneath the regions of a continental 
interior. Either directly or indirectly there must be a subcrustal 
transference going forward all the way between these distant 
regions; for example, from beneath the Mississippi and Colorado 
deltas to the fields of erosion in the Rocky Mountains, if a condi- 
tion of even approximate isostasy is to be maintained throughout. 
This does not mean of course that an individual ton of plastic 
rock is transferred a thousand miles to balance a ton of sediment. 
Each subcrustal unit may be transferred only a mile, but it involves 
a subsurface movement of matter all the way from the regions of 
sedimentation to the regions of erosion. 

Now this implies a continuous pressure-gradient, and even under 
the conception of great crustal weakness, a pressure-gradient which 
could fold the weak cover-rocks would be far higher than that 
needed for the movement of a continental glacier. Any large 
degree of viscous resistance in the zone of undertow would there- 
fore require, in order to initiate movement, an enormous defect 
of isostasy under the distant continental interior, an enormous 
excess under the marginal oceans. After a rejuvenative movement 
had started, it would be slow, the frictional and deformative resist- 
ances nearly balancing the deforming force. Therefore inertia of 
the moving mass could not carry it appreciably beyond the point 
where the moving force, weakened by loss of head, would just 
balance the resistances to further movement. It would be ex- 
pected, in consequence, that a residual pressure-difference would 
remain, even after a period of restorative isostatic movement. 
But an inspection of the map of New Method anomalies given in 
Part II, p. 153, does not show any such anomaly gradients as would 
comport with this expectation. A vast region of the continental 
interior extending from Lake Superior to the Rio Grande and west- 
ward to beyond the front ranges of the Rocky Mountains shows 
average positive anomalies, indicating an excess of matter, not a 


deficiency. To the westward is a broad region of average negative 
anomaly reaching a maximum at centers near the Pacific coast 
and no marked excess is shown near the mouths of the great 
rivers. Such a lack of regional relations would appear to show that 
the anomalies are due much more to local loads and irregularities 
upon and within the lithosphere, and to bowings due to great 
compressive movements unrelated to isostasy, rather than to the 
existence of an isostatic gradient leading from the ocean borders 
to the interior fields of great erosion. Therefore either the idea 
of strong viscous drag by undertow or else the very doctrine of 
isostasy — one or the other — must be abandoned. But it has been 
seen that if undertow exists in a comparatively plastic stratum, then 
that physical condition will cause it to be the bottom of the zone 
of compensation. Thus the application of every pertinent engineer- 
ing principle reduces the initial hypothesis of surface folding by 
isostatic undertow, and, especially by undertow within the zone of 
compensation, to an absurdity. 

Undertow restricted to a sphere of weakness — the asthenosphere. — 
All of this accumulative argument has not been advanced merely 
to show that a certain view is wrong. Rather has it been the inten- 
tion to prepare the ground for what would appear to be a sounder 
theory of the mode of maintenance of isostatic equilibrium. 

As for the basis of that theory, Schweydar, from the mathe- 
matical analysis of the measurement of the tides in the crust by 
means of the horizontal pendulum, has found that they are in 
accord with the assumption of the existence of a slightly plastic 
zone about 600 km. thick beneath a more rigid crust 120 km. thick. 1 
It would appear that the geodetic evidence of isostasy points 
also toward the existence of such a thick and somewhat plastic 
zone beneath the more rigid lithosphere. It gives no knowledge 
of the exact thickness or depth, but for convenience the figures 
given by Schweydar will be assumed. It is a matter of importance 
to note however that, although the quantitative limits are uncer- 
tain, the suggestions given both by the tides and by isostatic 

1 "Untersuchungen iiber die Gezeiten der festen Erde und die hypothetische 
Magmaschicht," VerSffentlichung des k. k. Preusz. geodat. Institutes, Neue Folge No. 54, 
Leipzig (191 2, B. G. Teubner). 



compensation point to a zone of weakness much deeper and thicker 
than the figures which have customarily been taken as a probable 
depth of origin of magmas. The latter however rests upon uncer- 
tain extrapolation, whereas the figures for the limits of the astheno- 
sphere, although of no exactness and perhaps 20 or 50 per cent 
from limits which finally may be chosen, have at least been deter- 
mined by more direct evidence. In such a thick shell of weakness, 
the readjustment, after an erosion cycle, of a continental interior 
to isostatic equilibrium would require but very little viscous shear 
and but little lateral movement. 



1.00a km. 


Fig. 14. — Diagrammatic vertical section of the crust, to show nature of undertow 
in the asthenosphere necessary to restore isostatic equilibrium in a positive interior 
continental area after a cycle of erosion. Effects of a vertical movement of 0.5 km. 
exaggerated 60 times. Asthenosphere grades into contiguous spheres and best limita- 
tions in depth are not known. 

To give quantitative visualization to this conclusion Fig. 14 is 
drawn. Suppose a plateau area i,oookm. wide in a continental 
interior to be separated from the region of sedimentary deposit 
by an intermediate region 1 ,000 km. across. Take a section 
1 km. wide through these regions. Let an erosion cycle cause the 
removal on the average of o . 5 km. of rock from this area to be 
deposited over an equal area of sea-bottom. Then, during an 
epoch of diastrophism, assume complete recovery of isostatic 
equilibrium by undertow in a sublithospheric zone of weakness 
600 km. thick. The vertical section of rock eroded is 500 sq. km. 
in area. As we have chosen a width of section of 1 km. we may 
also speak of this as the volume, 500 cu. km. To restore the mass 
of this column, 500 cu. km. must be added to it and flow past the 
vertical line which bounds it on the seaward side. As this zone of 


flow is 600 km. deep, the actual lateral movement, if all depths 
move equally, will be but o . 83 km., since o . 83 X 600 = 500. If the 
flowage is supposed to increase regularly from top and bottom to 
the middle the movement of the middle layer would be 1 . 66 km. 
A previously vertical line 600 miles long through this asthenosphere 
would then be bent at the middle by this amount and its two 
halves make angles of o°ic/ with the vertical. Each layer a kilo- 
meter thick would move horizontally 5.6m. with respect to each 
adjacent layer of kilometer thickness. These figures bring out the 
insignificant degree of the plastic deformation in such a deep zone 
which is needed to restore isostatic equilibrium, even for a large 
interior continental area after erosion amounting to two-thirds 
of the present average elevation of the North American continent. 

As a matter of fact the cross-section of the plastic deformation 
would not be a triangle, but a sinusoidal curve, so that the maximum 
linear flow for thickness of 600 km. would be between 0.83 and 
1 . 66 km. 

This illustration makes it clear that the isostatic rejuvenation 
of continental interiors as well as of the margins, which meets such 
grave difficulties under the hypothesis of a thin and shallow zone 
of isostatic undertow, is eliminated by adopting the hypothesis 
of a thick and plastic sublithospheric shell, such as has been found 
to be suggested by independent evidence. 

The idea of folding as a result of isostatic undertow definitely 
may be abandoned, but the absence of a notable isostatic gradient 
has some further significance. It is seen from Fig. 14 that if the 
fields of great erosion and deposition are within a few hundred 
kilometers of each other the rejuvenative undertow, under the 
laws of stress distribution in plastic bodies, would involve mostly 
a limited tract in the outer part of the asthenosphere; whereas, if 
the undertow must extend over distances of 1,000 km. or more, 
then the whole depth of the asthenosphere will become involved. 
The amount of stress-difference and of plastic shear per unit of 
volume may therefore be no greater in the one case than in the other. 
Especially, if the middle of the asthenosphere is its weakest part, 
a movement generated by areas large enough to involve the whole 
of this zone would go forward under less stress-difference per unit 


of area than for more local adjustments. The absence of a notable 
continental gradient is suggestive therefore of a deep zone of weak- 
ness, least resisting in its central portions, and of very marked 
plasticity in comparison with the rigidity of the lithosphere above. 
This does not involve, however, the conception of a truly fluid 
zone, but merely that of a comparatively plastic solid. 

The existence and nature of this zone of weakness is seen to 
enter vitally into the theory of isostasy and must of course bear 
with equal importance on other branches of terrestrial dynamics 
as well. It is proposed therefore to elevate it to equal rank with 
the other shells of the earth and to name it for that quality which, 
from the standpoint of diastrophism, is its most significant feature 
as compared to the zones above and below. This is its inability 
to resist stress-differences above a certain small limit. Its name, 
therefore, is the sphere of weakness — the asthenosphere. 

[To be continued]